Equivariant Neural Tangent Kernels

Abstract

Little is known about the training dynamics of equivariant neural networks, in particular how it compares to data augmented training of their non-equivariant counterparts. Recently, neural tangent kernels (NTKs) have emerged as a powerful tool to analytically study the training dynamics of wide neural networks. In this work, we take an important step towards a theoretical understanding of training dynamics of equivariant models by deriving neural tangent kernels for a broad class of equivariant architectures based on group convolutions. As a demonstration of the capabilities of our framework, we show an interesting relationship between data augmentation and group convolutional networks. Specifically, we prove that they share the same expected prediction at all training times and even off-manifold. In this sense, they have the same training dynamics. We demonstrate in numerical experiments that this still holds approximately for finite-width ensembles. By implementing equivariant NTKs for roto-translations in the plane ($G=C_{n}\ltimes\mathbb{R}^{2}$) and 3d rotations ($G=\mathrm{SO}(3)$), we show that equivariant NTKs outperform their non-equivariant counterparts as kernel predictors for histological image classification and quantum mechanical property prediction.